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Offloading Optimization and Bottleneck Analysisfor Mobile Cloud Computing

Di Han, Student Member, IEEE, Wei Chen, Senior Member, IEEE,Bo Bai, Senior Member, IEEE, Yuguang Fang, Fellow, IEEE

Abstract—Mobile cloud computing systems, or simply mobileclouds, have attracted tremendous attention because they allowmobile devices with limited computational resources to offloadcomplex computations. However, due to the channel uncertaintyand the complexity of a computation task, mobile computationoffloading may suffer from poor outage performance that theoffloaded task cannot be completed within the desired delay con-straint. Thus, how to efficiently identify and overcome the outagebottleneck, which could be used to optimize resource allocationschemes and improve the system performance effectively, is anopen problem. In this paper, we shall develop a unified frameworkthat minimizes the overall outage probability in various mobilecomputation offloading scenarios. More specifically, the outagebottleneck is defined and identified by adopting asymptoticanalysis, without any need of the accurate outage probabilitiesin both transmissions and computations. To overcome the outagebottleneck, resource pairing, matching, and allocation policies areinvestigated. Both theoretical analysis and numerical results showthat the outage bottleneck relies on not only the availability ofspectrum and computation resources but also the probabilitydistributions of computation complexities of the computationtasks.

Index Terms—Mobile cloud computing, computation offload-ing, outage bottleneck, power consumption.

I. INTRODUCTION

IN recent years, the demand for mobile devices to executeheavy applications with required delay constraints has

exhibited consistent growth. However, mobile devices areusually equipped with limited resources such as computationcapability [3] and battery power [4], due to the limitedform factor, e.g., physical size. The conflicting design con-sideration between resource-hungry applications and resource-constrained mobile devices hence poses a significant challengefor the future mobile computing development. To overcomethis challenge, mobile computation offloading has emerged asa potential technique, shifting complex computation tasks to

This work is supported in part by Beijing Natural Science Foundation(4191001), the National Science Foundation of China under Grant Nos.61671269 and 61621091, and the National Program for Special Supportfor Eminent Professionals of China (National 10,000 Talent Program). Thework of Y. Fang is supported in part by U.S. National Science Foundationunder grant CNS-1717736. The preliminary work has been presented at IEEEGlobalSIP’2016 [1] and IEEE GLOBECOM’2017 [2].

D. Han and W. Chen are with the Department of Electron-ic Engineering, Tsinghua University, Beijing 100084, China (e-mail:[email protected], [email protected]).

B. Bai was with the Department of Electronic Engineering, TsinghuaUniversity, Beijing 100084 China. He is now with Future Network The-ory Lab, Huawei Technologies Co.,Ltd., Shatin, Hong Kong (e-mail: [email protected]).

Y. Fang is with the Department of Electrical and Computer Engineering andUniversity of Florida, Gainesville, FL 32611, USA (e-mail: [email protected]).

more powerful computation facilities such as cloud serversor mobile cloud computation systems, and thus has attractedintensive from both academia and industries [5], [6] and [7].

Unfortunately, to offload a computation task, data may haveto be transmitted to the cloud, which may suffer from longlatency, and thus mobile clouds in proximity are preferred. Onone hand, if a mobile cloud is used, the computation resourcemay not be enough to complete the computation task on time.Furthermore, the power consumption on mobile devices is alsoan important factor to consider. Therefore, the computationmanagement is necessary in mobile cloud computing. Then,how to minimize the power consumption in a mobile devicewhile completing the required computation task on time ishighly challenging, but of paramount importance. This paperis to tackle this problem.

As mentioned, offloading a task from a mobile device to thecomputation resources in the cloud does reduce the workloadat the device, but will consume transmission power betweenthe device and cloud [4]. Due to the limited battery capacity ofmobile devices [8], energy efficiency becomes a critical issuein wireless transmission [9], and computation offloading [10]and [11]. Thus, the performance analysis of the end-to-endcomputation management between the mobile device and thecomputation resources is significantly important.

In this paper, we consider a mobile computation offloadingsystem composed of a mobile device connecting with multiplecomputation resources, simply called mobile cloud, throughmultiple wireless channels. During the computation offloading,each input task in a mobile device, ready for uploading to themobile cloud, could be divided into multiple subtasks and thenbe executed in parallel on different computation resources.However, the dynamic environment can cause additional prob-lems [12]. For example, the transmitted task may not reachthe computation resources, or the task executed on the servercannot be completed when it has to be returned to the mobiledevice. In addition, we assume that the input tasks are latency-sensitive. Thus, not only the transmission of each input taskand its output results but also the computation of the task mustbe completed within the required time latencies. Otherwise,the timeout will be triggered, resulting in task resubmission,which we simply state that the system is in outage state.The aforementioned outage event is composed of transmissionoutage and computation outage due to the uncertainty inwireless channels, and the lack of computation resource inthe mobile cloud. The outage probability is considered asthe performance metric. Both transmission and computationcapabilities could be improved by increasing the power of



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the system in order to lower the outage probability. However,the same level of improvement on outage performance mayrequire different levels of power consumption in transmissionand computation. The major factor that restricts the reductionof outage probability of the system even with power increaseis considered as the outage bottleneck, which could be used tooptimize resource allocation schemes to improve the systemperformance efficiently. Unfortunately, it is difficult to obtainexplicit expressions for the outage probabilities of transmissionand computation. Thus, how to identify and efficiently over-come the outage bottleneck remains as an open problem. In ourprevious work, we focus on this problem in a simple systemwith one channel and one computation resource [1] and thenextend the bottleneck into a system with multiple channels andcomputation resources [2]. However, a theoretical frameworkfor the bottleneck analysis which can be applied in moredifferent scenarios is still required.

The main contribution of this paper is to develop a unifiedframework from an asymptotic analysis perspective to analyzeand minimize the overall outage probability, and identifythe outage bottleneck of the considered system. Our unifiedframework jointly considers the task division and channelallocation in the computation offloading, and can be appliedin different offloading scenarios, e.g., distributed or centralizedcomputation, and with or without the information of the avail-able computation capabilities. In each considered offloadingscenario, the corresponding policy and algorithm are designedto minimize the overall outage probability. Then, by analyzingthe features of complexities of input task and backgroundtasks, necessary and sufficient conditions to identify the outagebottleneck of the system in each considered offloading scenariocan be obtained without knowing the accurate expressions ofthe outage probabilities of transmission and computation phas-es, which can be used to evaluate the outage performance anddesign an efficient strategy to improve the outage performancefor existing and future mobile computation offloading systems.

Throughout this paper, o(x) denotes the higher order in-finitesimal of x and max{a} denotes the maximal element inthe vector a. The symbol ∼ stands for equivalent infinites-imal. The indicator function 1 (a) is equal to 1 if a ≥ 1;otherwise, it is equal to 0. The rest of this paper is organizedas follows. The related work is summarized in Section II.Section III presents the system model and the frameworkfor the outage performance analysis. The analysis of outageperformance and outage bottleneck are given in Section IVand Section V. Numerical results are shown in Section VI toverify the theoretical results. Finally, Section VII concludes thepaper. For better illustration, most of the proofs for revealingour analytical results are put in the appendix and the mostimportant notations are listed in Table I.

II. RELATED WORK

There has been a lot of work on the energy efficiency ofthe computation offloading in the literature. In [10], Lin etal. proposed a framework for computation offloading basedon execution time and energy consumption, aiming to shortenresponse time and reduce energy consumption. In [13], Jiang et

TABLE ISUMMARY OF NOTATIONS

Symbol DefinitionC The set of the computation resourcesS The set of the wireless channels

CavaThe set of the computation resources received thewhole task after the upload phase

M , N The numbers of the computation resources and thewireless channels

Ts, T1, T2The lengths of the time slot, the upload phase, andthe download phase

lin, lout The data sizes of the input task and the output resultN0 The additive white Gaussian noise powerξc The power consumption of computationξt The power consumption of transmissionw The computation requirement of the input task

qiThe computation requirement of the background taskin the computation resource ci

C(ξc)The computation capability when the computationpower consumption is ξc

Hu(hiu

) The channel gain matrix (vector for the computationresource ci) in the upload phase

Hd

(hid

) The channel gain matrix (vector for the computationresource ci) in the download phase

A The channel allocation result in the download phasex The subtask division result in the computation phase

Riu, Ri

c, Rid

The successful ratios of computation, upload, anddownload phases in the computation resource ci

RThe successful offloading ratio, which is thepercentage of the task be offloaded successfully

pTypek(ξt, ξc)The overall outage probability in the scenario ofType k ∈ {1, 2, 3, 4}

pc,kout(M, ξc),pt,kout(N, ξt)

The computation and transmission outage probabilitypartitions in the scenario of Type k ∈ {1, 2, 3, 4}

ϕ1, φ1,ϕ2, φ2

Parameters used to identify the bottlenecks in thefour considered scenarios

al. designed a scheduling scheme for energy-efficient compu-tation offloading for multi-core mobile devices and developedan online algorithm based on Lyapunov optimization. In [14],Zhang et al. proposed a theoretical framework for mobile cloudcomputing under the stochastic wireless channel to determinethe optimal operational regions to save energy. A dynamiccomputation offloading policy was developed by You et al.in [15] and by Mao et al. in [11] to maximize the energysaving by considering microwave energy transfer and energyharvesting, respectively. In [14], Zhang et al. obtained closed-form solutions to decide the optimal application-executioncondition under which either the mobile execution or thecloud execution is more energy-efficient. In [16], Guo et al.integrated dynamic offloading with resource scheduling firstlyto minimize the joint energy consumption and applicationcompletion time under completion time deadline constraintand task-precedence requirement. The task In this work,we also focus on the energy efficiency of the computationoffloading through performance analysis in different scenarios.

More recently, various mobile computation offloading sys-tems have been considered. It was shown that energy efficiencycould be improved by jointly optimizing the allocation of com-putation and transmission resources [17]-[18]. In [17], Salmaniand Davidson jointly optimized the allocation of the compu-tational and radio resources to reduce energy consumption byexploiting the capabilities of the multiple access channel. In[19], Guo et al. considered the impact of task dependencies



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on computation completion time and then provided an energy-efficient dynamic offloading and resource scheduling policyto reduce energy consumption. In [20], Ti and Le studied thejoint computation offloading and resource allocation problemin the two-tier wireless heterogeneous network. In [21], Chenet al. aimed to find the local optimum decision on offloadingand allocation of computation and communication resources tominimize a weighted cost in a general mobile cloud computingsystem and an algorithm is given to obtain a local optimum.In [22], Wang et al. offered a joint energy minimization andresource allocation solution that could improve the systemperformance and save energy in a novel C-RAN architecturewith the mobile cloud co-located with the BBU in the cloudpool. In [18], Mao et al. dynamically managed the radio andcomputational resources to cope with the time-varying compu-tation demands and wireless fading channels. In this paper, wedevelop a unified framework to optimize the performance andimprove energy efficiency of various mobile cloud computingsystems from the perspective of outage probability.

Furthermore, by increasing the power consumption of bothtransmission and computation, the Quality-of-Service (QoS)of mobile users in mobile computation offloading systemscould be improved significantly [4][5]. However, the identicalincrease in power consumption of transmission and compu-tation might result in different improvement of the systemperformance and QoS. Hence, the transmission-computationpower consumption tradeoff is much more important andmeaningful, which has the potential to offer the most effectiveway to improve the system performance and has been studiedin [4] and our previous work [1] and [2]. Moreover, Ma et al.presented a survey of energy-efficient computation offloadingtechnologies and summarized that there is a tradeoff in energyconsumption between transmission and local computation in[23]. In this paper, to better utilize the resource in computationoffloading systems, we define the major limiting factor of thesystem as outage bottleneck.

III. SYSTEM MODEL

Consider a mobile computation offloading system that con-sists of one mobile device and M computation resources,through which the mobile device can offload the input tasksto the mobile cloud as shown in Fig. 1. The computationresources, whose set is denoted by C = {c1, . . . , cM}, aredistributed across different locations. Assume that the mobiledevice transmits with each computation resource throughN(N ≥ M) wireless channels, whose set is denoted byS = {s1, . . . , sN}. Due to the lack of computation capability,the input tasks of the mobile device need to be computedthrough computation offloading with the aid of the mobilecloud. In this way, a significantly improved computation expe-rience can be achieved. Moreover, all control operations of thecomputation offloading, i.e., transmissions and computations,are managed by a controller [12] in the considered system,which is in charge of processing the requests and providesmobile devices with the corresponding cloud services, e.g.,computation resources.

The procedure of the computation offloading can be dividedinto three phases: upload, computation, and download. Specif-

Fig. 1. Illustration of mobile computation offloading system with one mobiledevice connecting with multiple computation resources, where a controller isresponsible for all control plane decisions.

ically, each input task is set to be uploaded to the computationresource and then be computed in the mobile cloud. After thecomputation phase, the results of the task is transmitted backto the mobile device.

Let Ris denote the successful offloading ratio of the compu-tation resource ci, which is the percentage of one input taskthat can be offloaded successfully in the computation resourceci under the delay constraints. This successful offloading ratioRis is determined by Ric, Riu, and Rid, which are successfulratios of computation, upload, and download phases in thecomputation resource ci. In addition, if an input task cannot beuploaded completely, i.e., Riu < 1, then it cannot be computedin the computation resource ci, causing Ris = 0; otherwiseRis = min{Rid, Ric}. Therefore, Ris is given by

Ris = min{1(Riu), Rid, R

ic}. (1)

We derive next explicit expressions of Ric, Riu, and Rid withdifferent task division and channel allocation methods asfollows.

A. Concurrent Computation and Transmissions in Time Do-main

We consider a time-slotted system, in which the time isdivided into time slots with a fixed length Ts and is indexedby an integer k. Assume that each input task of the mobiledevice arrives at the beginning of the time slot, whose datasize (in bits) is a fixed value lin. Moreover, assume that thedata size of each task’s output result is also a fixed value lout.As depicted in Fig. 2, each time slot can be divided into twoparts: the upload phase T1 and the download phase T2, whereTs = T1 + T2.

Fig. 2. Time scheduling with concurrent computation and transmissions.

For the task arriving at the beginning of the k-th time slot,it is indexed as k-th task and uploaded from the mobile deviceto the mobile cloud in the upload phase of the k-th time slot,and then its output result is downloaded from the mobile cloud



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in the download phase of the (k+ 1)-th time slot. In this way,the wireless channels and computation resources can be wellutilized by concurrent computation and transmissions in timedomain, which has been used in our previous work [1] and[2]. Therefore, the delay constraints for upload, download, andcomputation phases are given by T1, T2, and Ts, respectively.

Since both the computation and transmission aspects play akey role, we next introduce the computation and transmissionmodels, respectively. Without loss of generality, we focus onthe offloading procedure and outage probability analysis forthe task arriving in the k-th time slot because the outage eventsof different tasks are independent and identically distributed.

B. Computation Model

The computation capability required by the task is measuredby the number of CPU cycles, denoted by w, which isa random variable with given probability density functionfw(x). In addition, there are also background tasks requiringcomputation capability in each computation resource. Let qidenote the computation capability required by backgroundtasks in the computation resource ci during the computationphase, which is also a random variable with given probabilitydensity function fq(x). Assume that w and qi (1 ≤ i ≤ M )are independent with each other.

The computation capability of each computation resourceis a finite value C, which is the total number of CPU cyclessupported by the computation resource during one computa-tion phase, i.e., one time slot Ts. Let fc denote the clockfrequency of the CPU in each computation resource. Thus,the computation capability of each computation resource isgiven by

C = fcTs. (2)

According to [24] and [25], the power consumption of com-putation can be expressed as

ξc = κfc3, (3)

where κ is the effective switched capacitance depending on thechip architecture of CPU. Then, the computation capability Ccan be rewritten as

C(ξc) = kcξc13 , (4)

where kc = Tsκ− 1

3 . Consequently, the successful ratio of thecomputation phase in the computation resource ci is given by

Ric(w, qi, ξc) =C(ξc)− qi

w. (5)

C. Transmission Model

The considered system is assumed to undergoRayleigh block-fading channel [26]. In each time-slot,the data link between the mobile device and one computationresource through each wireless channel has the independentand identical distribution (i.i.d.) in fading condition, whichare assumed to be circularly symmetric Gaussian distributionCN (0, 1) . The channel gain keeps fixed during one time-slot,while the channel gains are i.i.d. in different time slots.

Let hiu = [hi1u . . . , hiNu ]T and hid = [hi1d . . . , h

iNd ]T denote

the channel gain vectors for the computation resource ci inupload and download phases, where hiju and hijd are channelgains of upload and download links between the mobile deviceand computation resource ci over channel sj , respectively.Since computation resources are in different locations, it isreasonable to assume that the channels between the mobiledevice and different computation resources undergo indepen-dent fading. Therefore, both hiju and hijd , ∀1 ≤ i ≤ M ,are assumed to follow independent and identical distributionCN (0, 1). Furthermore, the channel gain matrices in uploadand download phases are denoted by Hu = [h1

u, . . . ,hMu ] and

Hd = [h1d, . . . ,h

Md ].

In the upload phase, to take full advantage of independentfading and maximize the diversity gain of transmission [27],the task is set to be transmitted to each computation resourcethrough the channel with maximal transmission capability,which is the point-to-multipoint communication. The broad-casting approach is not considered in the upload phase to avoidbroadcast storm when there are multiple mobile devices inrealistic scenarios. In this way, the mobile device can trans-mit the task to multiple computation resources through onewireless channel simultaneously and the transmission powerconsumption might be reduced compared to the broadcastingapproach. Thus, the successful ratio of the upload phase in thecomputation resource ci is given by

Riu(hiu, ξt) =T1 log(1 + |max{hiu}|2

ξtN0

)

lin, (6)

where ξt is the power consumption of transmission and N0 isthe power of the background noise.

In the download phase, there might have multiple compu-tation resources requiring channels to transmit results back tothe mobile device simultaneously. The channels occupied bythe computation resource ci in the download phase are denotedby ai = [ai1, . . . , aiN ]T , where aij = 1 if the channel sj isallocated to the computation resource ci; otherwise aij = 0.The channel allocation result can be represented by the matrixA = [a1, . . . ,aM ]. We notice that one channel cannot beallocated to multiple computation resources in order to avoidinterference. Thus, we have the constraint of channel allocationmethod, which is given by

M∑i=1

aij ≤ 1, ∀j = 1, . . . , N. (7)

Then the successful ratio of the download phase in thecomputation resource ci is given by

Rid(hid,ai, ξt) =

T2 log(1 + |maxsj∈S{aijhijd }|2

ξtN0

)

lout. (8)

D. Theoretical Framework for Outage Analysis

Assume that each input task can be divided into multiplesubtasks and then be computed in parallel on multiple com-putation resources. For the sake of simplicity, the separatedresults of all subtasks can be combined into a whole resultin the mobile device with negligible computation capability



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consumption [14] and [28], i.e., time delay can be ignored.The fractions of the original task in different computationresources are denoted by vector x = [x1, . . . , xM ], where xiis the fraction of the input task allocated to the computationresource ci. The subtask division method x can be designedby the controller, where the normalization condition holds andthus the sum of subtasks must cover the whole input task, i.e.,

M∑i=1

xi = 1, (9)

where 0 ≤ xi ≤ 1. Similar to the task model in [14] and[28], we assume the input task is uniformly splittable, i.e., afraction of xi of the original task has xilin input data bits,xilout output data bits, and requires xiw CPU cycles forprocessing. For some typical offloading applications, such asimage/video/voice recognition and file scanning, can satisfythis assumption [29] and then can be applied in the followingframework.

Based on the above analysis, the successful offloading ratioR, which is the percentage of one input task that can beoffloaded successfully, is given by

R(w,q,Hu,Hd,x,A, ξc, ξt) =M∑i=1

Ris =

M∑i=1

min{1(Riu(hiu, ξt)

), Rid(hid,ai, ξt), R

ic(w, qi, ξc), xi}.

(10)Thus, the system is in outage state if

R(w,q,Hu,Hd,x,A, ξc, ξt) < 1, (11)

which represents that the offloading of the input task cannotbe completed fully under the required delay constraints. FromEqs. (9)-(11), the system is in outage state if

1(Riu(hiu, ξt)

)< xi (12)

orRid(hid,ai, ξt) < xi (13)

orRic(w, qi, ξc)} < xi (14)

holds for any computation resource ci ∈ C. Then, a unifiedframework for outage performance analysis, where efficiencyand fairness are both taken into consideration, has already beenpresented. In this framework, x is a design parameter, whichcan be designed by the controller based on q and Hu afterthe upload phase. Next, A is also a design parameter, whichcan be determined based on x and Hd after the computationphase. The feedback of the above information will also causethe signaling overhead.

IV. OUTAGE PROBABILITY ANALYSIS

In this section, we focus on outage probability analysis indifferent offloading scenarios based on the unified frameworkin Eq. (10).

In the upload phase of each considered offloading scenario,the input task is set to be transmitted to all computation

resources over all available channels. If the input task cannotbe received by any one of the computation resources success-fully, the upload phase in this computation resource fails withprobability given by

poutbr (ξt) = Pr{Riu(hiu, ξt) < 1

}. (15)

Since the channel gains between the mobile device anddifferent computation resources over different channels areindependent with each other, the probability that the inputtask received by m (0 ≤ m ≤ M) computation resourcessuccessfully is given by

pupbr (m, ξt) =

(Mm

){1− poutbr (ξt)

}m {poutbr (ξt)

}(M−m).

(16)If no computation resource can receive the task successfully,i.e., m = 0, the computation offloading is considered to be afailure, i.e., in outage state, with probability given by

pupbr (0, ξt) ={poutbr (ξt)

}M. (17)

For each offloading scenario, the operation in the uploadphase is identical, while it might be different in computationand download phases, which are introduced next.

A. Classification of Computation Offloading Scenarios

Consider the condition that the whole task has been receivedby m (1 ≤ m ≤ M) computation resources after the uploadphase, whose set is denoted by Cava, where Cava ⊆ C and|Cava| = m. Thus, the task can be computed on at most mcomputation resources in parallel. Then, there are multipleoffloading scenarios should be considered, which are differentin the computation and download phases. Intuitively, theoffloading scenarios can be classified into two major categoriesaccording to either centralized or distributed computation,which are summarized in Table II.

TABLE IITWO COMPUTATION OFFLOADING CATEGORIES

Upload Computation DownloadPoint-to-multipoint Centralized Point-to-point

Point-to-multipoint Distributed Multiple access

Furthermore, the available computation capability of thecomputation resource ci for computation offloading is givenby C(ξt)− qi, which can be used by the controller to designthe subtask division method x. Hence, we define the valueC(ξt) − qi for Ci ∈ Cava as the computation capabilityinformation in the computation offloading. It should be noticedthat the value of C(ξc) and ξc are given, while qi is a randomvariable, which can be obtained through the feedback fromthe computation resource ci to the controller. It might notbe guaranteed in practical systems due to the constraint onsignaling overhead. Thus, another factor that should be con-sidered is whether the controller has computation capabilityinformation for all ci ∈ Cava. We consider two aforementionedfactors jointly and then four feasible scenarios are summarizedin Table III and clarified as follows:



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TABLE IIICLASSIFICATION OF FOUR TYPICAL SCENARIOS IN COMPUTATION

OFFLOADING

Computation Without computationcapability information

With computationcapability information

Centralized Type 1 Type 2Distributed Type 4 Type 3

1) Type 1: One computation resource ci∗ is selected fromCava randomly to compute the whole task. In this sce-nario, we have xi = 1 if i = i∗; otherwise xi = 0.

2) Type 2: One computation resource ci∗ with the maximalavailable computation capability, i.e., correspondinglythe minimal background task qi∗ , is selected from Cavato compute the whole task according to the given com-putation capability information. In this scenario, we havexi = 1 if i = i∗; otherwise xi = 0.

3) Type 3: The whole task is divided into m subtasks toeach available computation resources in Cava. Intuitively,the computation resource with high available computa-tion capability should be allocated to a relatively highportion of the whole task. Then the subtasks can becomputed in parallel on multiple computation resources.The task division method x can be obtained accordingto Algorithm 1 to minimize the outage probability of theoffloading.

4) Type 4: The whole task is divided into m subtasks withidentical size to each available computation resource inCava. In this scenario, we have xi = 1

m if ci ∈ Cava;otherwise xi = 0.

Next, we conduct the outage probability analysis with respectto the aforementioned scenarios.

B. Type 1: Centralized Computation without ComputationCapability Information

In this scenario, the controller selects one computationresource ci∗ from available computation resource set Cavarandomly. Hence the failure probability of computing is givenby

pcompType1(m, ξc) = Pr {w + qi > C(ξc)} , (18)

where qi∗ is replaced with qi since qi (1 ≤ i ≤ M) are i.i.d.random variables.

In the download phase, the result of the task is transmittedfrom ci∗ back to the mobile device, which is the point-to-point transmissions over N channels, i.e., ai∗ = [1, . . . , 1].Since xi∗ = 1, the failure probability of downloading is givenby

pdownType1(ξt) = Pr

{Ri

∗

d (hi∗

d ,ai∗ , ξt) < 1}. (19)

The outage events in the computation and download phasesare independent of each other since they depend on parametersand random variables of computations and transmissions sep-

arately. Hence the outage probability during the computationand download phases can be approximated by

pType1(m, ξt, ξc)

= pcompType1(m, ξc) + pdown

Type1(ξt)− pcompType1(m, ξc)p

downType1(ξt)

≈ pcompType1(m, ξc) + pdown

Type1(ξt),(20)

when pcompType1(m, ξc) and pdown

Type1(ξt) are sufficiently small,i.e., ξc and ξt are sufficiently large. In practical systems, theapproximation error in Eq. (20) is limited. This is simplydue to the fact that the outage probability in any practicalsystems must be small enough, or otherwise the QoS cannot beguaranteed, resulting in this scenario is unpractical. Therefore,the neglected part in approximation pcomp

Type1(m, ξc)pdownType1(ξt)

is much less than the overall outage probability of the systems,which can be neglected since the accuracy will not be reducedtoo much.

Then, From Eqs. (16), (17), and (20), the overall outageprobability in the scenario of Type 1 can be approximated by

pType1(ξt, ξc) ≈ pupbr (0, ξt)+M∑m=1

{pupbr (m, ξt)

(pcompType1(m, ξc) + pdown

Type1(ξt))}

.(21)

Thus, we obtain the following result.

Theorem 1. When ξc and ξt are sufficiently large, we have

pType1(ξt, ξc) ∼ k1ξ−Nt + pcompType1(M, ξc), (22)

where k1 ∈[N0

N2NR2 , N0N(2NR1M + 2NR2

)]with R1 =

linT1

and R2 = loutT2

.

Proof. See Appendix A.

C. Type 2: Centralized Computation with Computation Capa-bility Information

Given the computation capability information of all avail-able computation resources in Cava, the controller can selectthe computation resource ci∗ with minimal background tasksfrom Cava. Hence the failure probability of computing is givenby

pcompType2(m, ξc) = Pr

{maxci∈Cava

Ric(w, qi, ξc) < xi

}= Pr

{w + min

i=1,...,mqi > C(ξc)

},

(23)

which is different from that in the scenario of Type 2 givenin Eq. (18). During the offloading, the operation in thecomputation phase in the scenarios of Type 1 and Type 2are different, while that in the upload and download phasesare identical. Thus, similar to the proof of Theorem 1, we canobtain the following result for the overall outage probabilitypType2(ξt, ξc) in this scenario.



where N0N2NR2 ≤ k2 ≤ N0

N(2NR1M + 2NR2

)with R1 =

linT1

and R2 = loutT2

.



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D. Type 3: Distributed Computation with Computation Capa-bility Information

Given the computation capability information of all com-putation resources in Cava, the computation capability canbe fully exploited by finding the optimal subtask divisionmethod x∗ at the controller. In the ideal case, the taskcannot be completed only if the sum of available computationcapabilities of all computation resources in Cava cannot satisfythe computation requirement of the task. From Eqs. (5) and(14), the probability of the task cannot be completed duringthe computation phase in above ideal case is given by


{w >

m∑i=1

max{C(ξc)− qi, 0}

}, (25)

where max{C(ξc) − qi, 0} represents the available computa-tion capability in computation resource ci ∈ Cava. Since thecomputation resource will be fully loaded when C(ξc) ≤ qi,the available computation capability of each computationresource cannot be negative. To obtain the optimal subtaskdivision method x∗ minimizing the overall outage probabilityof offloading, an algorithm with low computation complexityis proposed in Algorithm 1. Based on this, we have thefollowing theorem.

Algorithm 1 Subtask Division Algorithm1: initialize the sets Cava and x = 0.2: repeat3: m = |Cava|.4: qimax = max

ci∈Cava{qi} and imax = arg max

ci∈Cava{qi}.

5: if C(ξc)− qimax >wm , then ximax = 1

m .6: else ximax = C(ξc)−qimax

w , w = (1− ximax)w anddelete cimax from Cava.

7: until Cava = ∅.8: output x∗ = x, then stop.

Theorem 3. By implementing the subtask division method x∗

obtained from Algorithm 1, the overall outage probability ofcomputation offloading can be minimized if

∑ci∈Cava

xi = 1.

Otherwise if∑

ci∈Cavaxi < 1, the offloading will be in outage in

the computation phase for any subtask division methods.

Proof. See Appendix B.

Then, in the download phase, the result of each subtask istransmitted from the corresponding computation resource inCava back to the mobile device. From Eqs. (9) and (14), thefailure probability of downloading subtask from ci throughchannel sj is given by

pdown,iType3 (xi, ξt) = Pr

{T2 log(1 + |hijd |2

ξtN0

)

lout< xi

}= Pr

{Rid(hid, [1, 0, . . . , 0], ξt) < xi

},

(26)

because the channel gains hi1d and hijd , j = 2, . . . , N , are i.i.d.random variables.

The download phase in this scenario is to conduct multipleaccess transmissions of m computation resources over Nchannels (m ≤M ≤ N), and optimal channel allocation canbe formulated as a matching problem in the random bipartitegraph (RBG) [30] and then be solved by the bipartite graphmatching. The RBG model can be constructed in the followingsteps.

First, all the vertices in one partition class are used to rep-resent the available computation resources ci ∈ Cava, and thevertices in another partition class are used to represent all thechannels sn ∈ S. Only if the result of the subtask in ci ∈ Cavacan be downloaded through channel sn ∈ S successfully canwe add an edge between the corresponding vertices in theRBG sample. A sample of this RBG model with m = 4 andN = 6 is shown in Fig. 3. By applying the RBG model,the optimal channel allocation can be obtained according toR2HK Algorithm with the complexity of O(N2.5) [30].

Fig. 3. A sample of RBG model with 4 available computation resources and6 wireless channels.

Let pdowntype3(m, ξt) denote the failure probability of down-

loading in the scenario of Type 3, which is the probabilitythat not all results can be downloaded successfully with thesubtask division method x∗ according to Algorithm 1. Thefollowing theorem can be obtained.

Theorem 4. For sufficiently large ξt, the failure probability ofdownloading pdown

type3(m, ξt) under channel allocation schemebased on R2HK algorithm is given by

pdowntype3(m, ξt) =

∑ci∈Cava

{pdown,1Type3 (xi, ξt)

}N+ a0(m)·

N∏

ci∈Cava

pdown,1Type3 (xi, ξt) + o

( ∑ci∈Cava

{pdown,1Type3 (xi, ξt)

}N),

(27)where

a0(m) =

{0, m < N

1, m = N.(28)

Proof. See Appendix C.

From this Theorem, we can easily obtain the followingresult.

Corollary 1. For sufficiently large xt, pdowntype3(m, ξt) can be

approximated by

pdowntype3(m, ξt) ≈ kdξt−N , (29)



8

where kd ∈[m2xminR2N , (m+N)2xmaxR2N

]· N0

Nξ−Nt ,xmax = max

ci∈Cavaxi, and xmin = min

ci∈Cavaxi.

Proof. See Appendix D.

When ξc and ξt are sufficiently large, the overall outageprobability in this scenario can be approximated by

pType3(ξt, ξc) ≈ pupbr (0, ξt)

+M∑m=1

{pupbr (m, ξt)


Type3(m, ξt))}

,

(30)and we can obtain the following theorem.



where k3 ∈[kd, N0

N2NR1M + kd]

with R1 = linT1

and R2 =loutT2

.

E. Type 4: Distributed Computation without ComputationCapability Information

Without known computation capability information of com-putation resources ci ∈ Cava in the controller, a simple butreasonable method for the controller is to divide the wholetask into subtasks with identical size and then the subtasksare computed in each computation resource in the availablecomputation resource set, i.e., xi = 1

m for ci ∈ Cava. FromEqs. (5) and (14), the failure probability of computing is givenby


{minci∈Cava

Rci (w, qi, ξc) < xi

}= Pr

{w

m+ maxi=1,...,m

qi > C(ξc)

}.

(32)

In contrast to this scenario, the scenario of Type 3 without theconstraint of equal task division is more general. Thus, thetask division method in the scenario of Type 4 can be viewedas one case of that in the scenario of Type 3. By substitutingxi = 1

m for ci ∈ Cava in Eq. (26), the failure probability ofdownloading subtask from ci through channel sj is given by

pdownType4(ξt) = pdown,i

Type3 (1

m, ξt)

= Pr

{Rdi (hd

i , [1, 0, . . . , 0], ξt) <1

m

}.

(33)

Then, from Eq. (27) in Theorem 4, the failure probability ofdownloading is given by

pdowntype4(m, ξt) = b(m)

{pdownType4(ξt)

}N+ o

(pdownType4(ξt)

N),

(34)where b(m) = a(m) + 1. When ξc and ξt are sufficientlylarge, the overall outage probability in this scenario can beapproximated by

pType4(ξt, ξc) ≈ pupbr (0, ξt)

+M∑m=1

{pupbr (m, ξt)


Type4(m, ξt))}

,

(35)

and the following theorem can be obtained.



where k4 ∈[b(M)MN0

Nλ2, b(M)MN0N (λ1 + λ2)

]with

λ1 = 2NlinT1 and λ2 = 2

NlinT2 .

V. OUTAGE BOTTLENECK ANALYSIS

According to Theorems 1, 2, 5, and 6, the outage prob-abilities with high power consumption of computation andtransmission in all considered scenarios can be divided intotwo parts, which only depend on transmission parameters Nand ξt, and computation parameters M and ξc, respectively.Thus, we define these two partitions as the computation andtransmission outage probability partitions. For the sake ofsimplicity, we set the constant coefficients in transmissionoutage probability in Theorems 1, 2, 5 and 6, i.e., k1, k2, k3and k4, to 1. Therefore, we have the following definition.

Definition 1. The computation and transmission outage prob-ability partitions of four considered scenarios are defined by{

pc,kout(M, ξc) = pcompTypek(M, ξc),

pt,kout(N, ξt) = ξ−Nt ,(37)

where k ∈ {1, 2, 3, 4} that correspond to Type 1, 2, 3, and 4scenarios.

Then, for given M and N , the power consumptions withpc,kout(M, ξc) = pt,kout(N, ξt) = ε can be rewritten as ξc(ε) = pc,kout

−1(ε) = fk(ε),

ξt(ε) = pt,kout−1

(ε) = gk(ε),(38)

where p−1(x) is the inverse function of p(x). The major part oftotal power consumption as ε→ 0 is considered as the outagebottleneck of the mobile computation offloading system, whichis the major limiting factor to the decrease of outage probabili-ty of the system by increasing total power consumption. Then,according to Eq. (38), the outage bottleneck of the computationoffloading can be found by comparing power consumptionratio

hk(ε) =ξc(ε)

ξt(ε)=fk(ε)

gk(ε)(39)

with ε → 0 and k ∈ {1, 2, 3, 4}, which is summarized in thefollowing definitions.

Definition 2. Define1

Ξk = limε→0

hk(ε) = limε→0

fk(ε)

gk(ε)= limξc→∞

ξc

gk(fk−1(ξc))

, (40)

where k ∈ {1, 2, 3, 4}. If Ξk = 0 or ∞, the outage bot-tleneck of corresponding offloading scenario is either trans-mission or computation capability, respectively. Otherwise, ifΞk ∈ (0,∞), the outage bottleneck is both transmission andcomputation capability.

1In this paper, we do not consider the scenario that the limitation limε→0

hk(ε)

does no exist.



9

The rest of this paper focuses on the analysis of the featuresand acquisition method of Ξk to obtain outage bottleneck,which are summarized in the following theorems.

Theorem 7. Ξk defined in Eq. (40) can be rewritten as

Ξk = W limx→∞

lk(x), (41)

where k ∈ {1, 2, 4}, W is a constant satisfying W =1

kc3{3N}1N∈ [0,∞] , and lk(x) is given by

lk(x) =

{fuk(x)

x−(3N+1)

} 1N

(42)

with fuk(t) being the probability density function of uk, whichis given by

uk =

w + qi, k = 1;

w + mini=1,...,M

qi, k = 2;

w

M+ maxi=1,...,M

qi, k = 4.

(43)

Proof. See Appendix E.

According to Theorem 7, the outage bottleneck in thescenarios of Type 1, 2, and 4 can be obtained by analyzinglimx→∞

lk(x). However, it is sometimes difficult to obtain explicitexpression of lk(x). To solve this problem and obtain the out-age bottleneck more efficiently, we first analyze distributionsof w, qi and uk.

Definition 3. Defineϕ1 = limx→∞

x3N+1fw(x);

φ1 = limx→∞

x3N+1fq(x),(44)

where fw(x) and fq(x) are the probability density functionsof w and qi, respectively.

Theorem 8. In the scenarios of Type 1 and Type 4, ifmax{ϕ1, φ1} = 0 or ∞, the outage bottleneck of this systemis either transmission or computation capability, respectively.Otherwise, if max{ϕ1, φ1} ∈ (0,∞), the outage bottleneck isboth transmission and computation capability.

Proof. See Appendix F.

Similar to Definition 3 and Theorem 8 for the scenarios ofType 1 and Type 4, we have the following definitions for thescenarios of Type 2 and Type 3.

Definition 4. Defineϕ2 = limx→∞

x3N+1fw(x);

φ2 = limx→∞

x3NM +1fq(x),

(45)

where fw(x) and fq(x) are the probability density functionsof w and qi, respectively.

Theorem 9. In the scenarios of Type 2 and 3, ifmax{ϕ2, φ2} = 0 or ∞, the outage bottleneck of this systemis either transmission or computation capability, respectively.Otherwise, if max{ϕ2, φ2} ∈ (0,∞), the outage bottleneck isboth transmission and computation capability.

Proof. For the scenario of Type 2, the proof is similar to thatfor the scenario of Type 1 in Appendix F. In addition, for thescenario of Type 3, the proof is given in Appendix G.

Remark 1. According to [31], the number of required CPUcycles w depends on the input data size lin. For a given inputdata size lin, the number of required CPU cycles can be derivedas

w = Llin, (46)

where L indicates the number of CPU cycles per bit, whichcan be modeled by a Gamma distribution with shape parameterα and scale parameter β in [31]. In this case, we have

fw(x) =1

βΓ(α)lin(x

βlin)α−1e

− xβlin , x > 0, (47)

Then, ϕ1 and ϕ2 can be calculated as

ϕ1 = ϕ2 = limt→∞

x(3N+1)fw(x) = 0, ∀N ≥ 1. (48)

According to Theorem 8 and Theorem 9, the outage bottleneckof four considered offloading scenarios only depends on φ1or φ2. If qi also follows a Gamma distribution, i.e., φ1 =φ2 = 0, the outage bottlenecks of four offloading scenariosare transmission capability.Remark 2. If w and qi are light-tailed distributions as definedin [32], we have ϕ1 = φ1 = 0 and ϕ2 = φ2 = 0, and theoutage bottlenecks of four considered offloading scenarios aretransmission capability. However, if w and qi are heavy-taileddistribution as defined in [32], the outage bottlenecks can alsobe obtained according to Theorem 8 and Theorem 9.

VI. NUMERICAL RESULTS

In this section, we validate the theoretical analysis viaMonte-Carlo simulation results. Throughout this section, weset kc = 1, the additive white Gaussian noise N0 = 1, datasize of the input task lin = 5, and data size of the outputresult lout = 0.5. The lengths of the time slot, the uploadphase, and the download phase are assumed to be normalized,i.e., T1 = 1, T2 = 1, and Ts = 2. In addition, the channelgains hiju and hijd are assumed to follow independent andidentical distribution CN (0, 1). Moreover, the distributions ofcomputation requirements w and qi are given by

1) Case 1: w and qi follow the identical independent Gam-ma distribution with α = β = 1, i.e., the exponentialdistribution with parameter λ = 1.

2) Case 2: w and qi follow the independent power-lawdistribution, whose probability density functions arefw(x) = 10

x11 and fq(x) = 5x6 for x ≥ 1, respectively.

The controller collects the above information and schedulesthe offloading based on the proposed approach. Then, notonly the overall outage probability of offloading, but alsothe computation and transmission outage probability partitionsdefined in Eq. (37) can be obtained.

For two cases, the ϕ1, φ1, ϕ2, and φ2 defined in Definitions3 and 4 are summarized in Table IV. According to Theorem8 and Theorem 9, we can further obtain the outage bottleneckin four considered scenarios with case 1 and case 2, which issummarized in Table V.



10

TABLE IVϕ1/2 AND φ1/2 IN TWO CASES

fw , fqCase 1:

exponentialCase 2:

power-lawϕ1 0 0 (N ≤ 3) and ∞ (N ≥ 4)ϕ2 0 0 (N ≤ 3) and ∞ (N ≥ 4)φ1 0 0 (N ≤ 1) and ∞ (N ≥ 2)φ2 0 0 (3N < 5M ) and ∞ (3N > 5M )

TABLE VOUTAGE BOTTLENECKS IN TWO CASES

Scenario Case 1:exponential

Case 2:power-law

Type 1 Transmission Transmission (N ≤ 1)and computation (N ≥ 2)

Type 2 Transmission Transmission (N ≤ 3 and 3N < 5M )and computation (in others)

Type 3 Transmission Transmission (N ≤ 3 and 3N < 5M )and computation (in others)

Type 4 Transmission Transmission (N ≤ 1)and computation (N ≥ 2)

A. Overall outage probabilities in different scenarios

The overall outage probabilities in four considered sce-narios are given in Figs. 4 and 5, versus different powerconsumptions of computation and transmission, respectively.With the increase of the power consumptions of computationand transmission, the overall outage probabilities decreasein all considered scenarios. In addition, for the scenarios ofcentralized computation, the scenario of Type 2 always hasa better outage performance than the scenario of Type 1.Similarly, for the scenarios of distributed computation, theoverall outage probability in the scenario of Type 3 alwaysbelow that in the scenario of Type 4. Hence it shows that theconsidered system can achieve a better outage performancewith the help of the computation information.

4 8 12Power Consumption of Computation: ξc

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Ove

rall

Out

age

Pro

babl

ity

Type1's scenarioType2's scenarioType3's scenarioType4's scenario

Fig. 4. Overall outage probabilities with ξt = 15.

B. One simple system with M = 1

We consider a simple scenario that M = 1 firstly. Sincethere is only one computation resource in the system, allscenarios will degrade into the scenario of Type 1.

Fig. 6 indicates the power consumption ratio ξcξt

definedin Eq. (39) with the assurance that ptk(ξt) = pck(ξc), k ∈{1, 2, 3, 4}, with the increase of ξc. By increasing the number

5 10 15Power Consumption of Transmission: ξt

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Ove

rall

Out

age

Pro

babl

ity

Type1's scenarioType2's scenarioType3's scenarioType4's scenario

Fig. 5. Overall outage probabilities with ξc = 12.

of channels N from 1 to 4, the power consumption ratio ofcase 1 always tends to 0, while that of case 2 tends to∞ whenN ≥ 2. It shows that the outage bottleneck for case 1 when1 ≤ N ≤ 4 and for case 2 when N = 1 is transmissioncapability, while for case 2 with N ≥ 2 is computationcapability, where theoretical results and the simulation resultsmatch perfectly well.

101 102 103 104 105

Power Consumption of Computation: ξc

10-15

10-10

10-5

100

105

PowerConsumptionRatio:

ξ c ξ t

Reference Line:ξc

ξt= 1

Case 1 with N=1

Case 1 with N=2

Case 1 with N=3

Case 1 with N=4

Case 2 with N=1

Case 2 with N=2

Case 2 with N=3

Case 2 with N=4

Bottleneck is transmission

Bottleneck is computation

Fig. 6. The power consumption ratio with M = 1, i.e., Type 1’s scenario.

Next we consider a system with multiple wireless channelsand computation resources.

C. Centralized Computation with and without ComputationInformation

101 102 103 104 105


10-2

10-1

100

101

102

103


ξ c ξ t

Reference Line:ξc

ξt= 1

Case 1 with L=1

Case 1 with L=2

Case 1 with L=3

Case 2 with L=1

Case 2 with L=2

Case 2 with L=3



Fig. 7. The power consumption ratio in Type 1’s scenario.

Consider the scenarios of centralized computation and thenumber of channels N is 3, where the scenario of Type 1 has



11

101 102 103 104 105


10-2

10-1

100

101

102

103


ξ c ξ t

Reference Line:ξc

ξt= 1

Case 1 with L=1

Case 1 with L=2

Case 1 with L=3

Case 2 with L=1

Case 2 with L=2

Case 2 with L=3




been presented in the previous [2] and that of Type 2 has beenpresented in Fig. 7. It shows that the power consumption ratioξcξt

defined in Eq. (40) with the increase of ξc in the scenariosof Type 1 and Type 2. As shown in Fig. 7, by increasing thenumber of computation resources M from 1 to 3, the powerconsumption ratios in the scenario of Type 1 always tends to 0for case 1, and ∞ for case 2. Moreover, Fig. 8 shows that, inthe scenario of Type 2, the power consumption ratio ξc

ξtalways

tends to 0 for case 1, while that of case 2 tends to 0 whenM ≥ 2 and to ∞ when M = 1 with the increase of ξc. Thus,with the known of computation capability information, theoutage bottleneck in the scenario of Type 2 can be transformedfrom computation to transmission by increasing the number ofcomputation resources. The numerical results in Figs. 7 and8 match well with the theoretical results in Theorem 8 andTheorem 9.

D. Distributed Computation with and without ComputationInformation

101 102 103 104 105


10-15

10-10

10-5

100

105


ξ c ξ t

Reference Line:ξc

ξt= 1

Case 1 with M=1

Case 1 with M=2

Case 1 with M=3

Case 2 with M=1

Case 2 with M=2

Case 2 with M=3




Consider the two scenarios of distributed computation andthe number of channels N is 3. Figs. 9 and 10 show the powerconsumption ratio ξc

ξtdefined in Eq. (40) with the increase of

ξc in the scenarios of Type 3 and Type 4. As shown in Fig. 9,by increasing the number of computation resources M from 1to 3, the power consumption ratio of case 1 always tends to 0,while that of case 2 with M ≤ 1 tends to∞ with the increaseof ξc in the scenario of Type 3. In addition, Fig. 10 shows

101 102 103 104 105


10-2

100

102

104

106

Reference Line:ξc

ξt= 1

Case 1 with M=1

Case 1 with M=2

Case 1 with M=3

Case 2 with M=1

Case 2 with M=2

Case 2 with M=3




that the power consumption ratios of case 1 and case 2 inthe scenario of Type 4 always tend to 0 and ∞, respectively.With the known of computation capability information, theoutage bottleneck in the scenario of Type 3 can be transformedfrom computation to transmission by increasing the number ofcomputation resources. The numerical results in Figs. 9 and10 also match with the theoretical results in Theorem 8 andTheorem 9.

VII. CONCLUSIONS

In this paper, we have developed a unified frameworkto analyze the outage probability and outage bottleneck formobile computation offloading systems. Offloading scenarioswith different task division and channel allocation methods areconsidered. In each considered scenario, the overall outageprobability can be divided into two parts corresponding totransmission and computation. The outage bottleneck com-pletely determined by the power consumption ratio with highpower consumption. By analyzing the computation complex-ities of tasks and bounding the outage probability, we havediscovered the conditions for outage bottleneck. The theoreti-cal results can be used to evaluate the outage performance andbottleneck for existing and future mobile computation offload-ing systems and improve outage performance in an efficientway. Important future topics include the extension to othersystem models, e.g., with local computing, the implementationof the distributed control, scheduling that takes into accountthe order of tasks and more influencing factors, as well as,practical bottleneck evaluation based on real data.

APPENDIX APROOF OF THEOREM 1

Proof. According to Eqs. (15) and (19), we havepoutbr (ξt) =

(1− e−2

R1 N0ξt

)N;

pdownType1(ξt) =

(1− e−2

R2 N0ξt

)N,

(49)

When ξc and ξt are sufficiently large, Eq. (49) can be rewrittenas

poutbr (ξt) ≈(

2R1N0

ξt

)Nand pdown

Type1(ξt) ≈(

2R2N0

ξt

)N.

(50)



12

Moreover, we have

pcompType1(m, ξc) ∈ [0, 1],∀m = 1, . . . ,M. (51)

Thus substituting Eqs. (50) and (51) into Eq. (21), we provethe theorem in Eq. (52) as follows:

APPENDIX BPROOF OF THEOREM 3

Proof. According to Eqs. (8) and (26), when ξt is sufficientlylarge, we have

pdown,iType3 (xi, ξt) ≈ 2xiR2N0ξt

−1. (53)

Then, by taking Eq. (53) into Eq. (27), we have

pdowntype3(m, ξt)

≈(N0

ξt

)N ∑ci∈Cava

2NxiR2 +

(N0

ξt

)ma0(m)N

∏ci∈Cava

2xiR2

=

(N0

ξt

)N ∑ci∈Cava

(2NR2

)xi+

(N0

ξt

)ma0(m)N2R2 ,

(54)

wherem∑i=1

xi = 1. Therefore, minimizing pdowntype3(m, ξt) is

equivalent to minimizing∑ci∈Cava

(2NR2

)xi=

∑ci∈Cava

αxi , (55)

where α = 2NR2 > 1. Then, by considering a case that x1 =β1 and x2 = β2 with β1 < β2, we have

(αβ1 + αβ2)− 2αβ1+β2

2 > 0, (56)

according to Basic Inequality that a + b > 2√ab with

a, b > 0. For the purpose of minimizing the value in Eq. (55),(x1, x2) = (β1+β2

2 , β1+β2

2 ) is better than (β1, β2) when m =2. According to the above proof by contradiction, the optimalx minimizing Eq. (55) is xi = 1

m for ci ∈ Cava; otherwisexi = 0. However, according to Eq. (14), to avoid outage inthe computation phase, the condition xi ≤ max{C(ξc)−qi,0}

w isrequired for all ci ∈ Cava, which might not always hold.

Furthermore, we should allocate the subtask to the compu-tation resource with qimax = max

ci∈Cavaqi firstly. The fraction

of task ximax in cimax is the lower bound among that ofall resources in Cava. From Eqs. (5) and (14), we have theupper bound of ximax is C(ξc)−qimax

w , which is the fractionthat cimax can support while guaranteeing no outage during thecomputation phase. Similar to the contradiction in Eq. (56),we could prove that (x1, x2) = (β − ε, ε) is better than(β, 0), where ε is a small constant. Thus, we set ximax =

min{ 1m ,

C(ξc)−qimax

w } to minimize pdowntype3(m, ξt) while avoid

outage in the computation phase if at all possible. After thisoperation, the unallocated fraction of the whole task is updatedto 1− ximax and the set of computation resources waiting toallocate the subtasks is Cava\cimax. Then we repeat the aboveoperation until all resources have already be assigned to theirsubtasks, i.e., Cava = ∅. The above operations are summarizedin Algorithm 1.

APPENDIX CPROOF OF THEOREM 4

Proof. If m < N , according to Lemma 6 in [30], there is onesubtask that cannot be downloaded successfully only when thesubtask cannot be downloaded through any channel in S. Thatis, this subtask is an isolated vertex in the RBG sample.

This event can occur in the subtask in any computationresources Cava. If m = N , according to Lemma 6 in [30], thereis one subtask that cannot be downloaded successfully onlywhen the subtask cannot be downloaded through any channelin S or all subtasks cannot be downloaded through one channelin S. That is, this subtask or one channel is an isolated vertexin the RBG sample. This event can occur in the subtask inany computation resources Cava and any channel in S. Thus,similar to the proof of Theorem 1 in [30], we obtain the proofof the theorem.

APPENDIX DPROOF OF COROLLARY 1

Proof. According to Eq. (53), for given ξt, we can find thatpdown,1Type3 (xi, ξt) is monotonously increasing in the fraction of

the whole task xi. Therefore, pdown,1Type3 (xi, ξt) can be bounded

by

pdown,iType3 (xmin, ξt) ≤ pdown,i

Type3 (xi, ξt) ≤ pdown,iType3 (xmax, ξt).

(57)Then, by taking Eq. (53) in Eq. (27), we have

pdownType3(m, ξt)

≈m∑i=1

(2xiR2

N0

ξt

)N+ a0(m)N

m∏i=1

(2xiR2

N0

ξt

).

(58)

Hence, by taking Eq. (58) into Eq. (57), we have

pdownType3(m, ξt) ≥ m2xminR2NN0

Nξ−Nt , (59)

and

pdownType3(m, ξt) ≤ (m+N)2xmaxR2NN0

Nξ−Nt . (60)

APPENDIX EPROOF OF THEOREM 7

Proof. According to Eqs. (18), (23) and (32), we have

gk(fk−1(ξc)) =

1

Pr{uk > kcx13 }

1N

. (61)

For any k ∈ {1, 2, 4}, by taking Eq. (61) into Eq. (40) andusing L’Hopital’s rule, we have

Ξk = limx→0

hk(x) = limx→∞

x

gk(fk−1(x))

= limx→∞

{Pr{uk > kcx

13 }

x−N

} 1N

= limx→∞

1

kc3

{Pr{uk > x}

x−3N

} 1N

= limx→∞

1

kc3

{fuk(x)

3Nx−(3N+1)

} 1N

= W limx→∞

lk(x).

(62)



13

pType1(ξt, ξc)

∼(

2R1N0

ξt

)NL+

M∑m=1

{(Mm

){1−

(2R1N0

ξt

)N}m(2R1N0

ξt

)N(M−m){pcompType1(m, ξt) +

(2R2N0

ξt

)N}}

∼M∑m=1

{(Mm

)(2R1N0

ξt

)N(M−m){pcompType1(m, ξt) +

(2R2N0

ξt

)N}}

∼ pcompType1(M, ξt) +

(2R2N0

ξt

)N+M

(2R1N0

ξt

)NpcompType1(M − 1, ξt)

∈[pcompType1(M, ξt) +

(N0

N2NR2)ξ−Nt , pcomp

Type1(M, ξt) +N0N(2NR1M + 2NR2

)ξ−Nt

].

(52)

APPENDIX FPROOF OF THEOREM 8

Proof. Since both w and qi are non-negative random variablesand independent of each other, we have u1 = w+ qi is largerthan w and qi, and Pr {u1 > x} ≥ Pr {w > x} ;

Pr {u1 > x} ≥ Pr {qi > x} ;Pr {u1 > x} ≤ Pr

{w > x

2

}+ Pr

{qi >

x2

}.

(63)

By substituting Eq. (63) into Ξ1 in Eq. (41), we have

Ξ1 ≥ limx→∞

{Pr

{w≥kcx

13

}x−N

} 1N

;

Ξ1 ≥ limx→∞

{Pr

{qi≥kcx

13

}x−N

} 1N

;

Ξ1 ≤ limx→∞

Pr

{w≥ kcx

13

213

}+Pr

{q≥ kcx

13

213

}x−N

1N

.

(64)

Thus, by using L’Hopital’s rule, the inequalities in Eq. (64)can be rewritten as

Ξ1 ≥ limx→∞

1kc3

{fw(x)

3Nx−(3N+1)

} 1N

= k−3c (3N)−1N ϕ1

1N ;

Ξ1 ≥ limx→∞

1kc3

{fq(x)

3Nx−(3N+1)

}MN

= k−3c (3N)−1N φ1

1N ;

Ξ1 ≤ 8k−3c

((3N)−1ϕ1 + (3N)−1φ1

) 1N .

(65)Therefore, Ξ1 can be bounded as follows

k−3c θ1N ≤ Ξ1 ≤ 8k−3c (2θ)

1N , (66)

where θ = 13N max{ϕ1, φ1}, which shows that the upper

and lower bounds of Ξ1 can be determined by ϕ1 and φ1.Moreover, Ξ4 can be bounded by similar method.

APPENDIX GPROOF OF THEOREM 11

Proof. It is easy to find that

max {C(ξc)− q, 0} ≤∑

ci∈Cava

max{C(ξc)− qi, 0}

≤ mmax {C(ξc)− q, 0}(67)

where |Cava| = m and q = minci∈Cava

qi. Then, by taking Eq. (67)

into Eq. (25), we have


{w >

∑ci∈Cava

max(C(ξc)− qi, 0)

}∈[Pr{wm

+ q > C(ξc)},Pr {w + q > C(ξc)}

].

(68)Since w and qi are non-negative random variables and

independent of each other, we havepcompType3(M, ξc) ≥ Pr

{wM > C(ξc)

};

pcompType3(M, ξc) ≥ Pr {q > C(ξc)} ;

pcompType3(M, ξc) ≤ Pr

{w > C(ξc)

2

}+ Pr

{q > C(ξc)

2

}.

(69)By substituting Eq. (69) into Ξ3 in Eq. (40), we have

Ξ3 ≥ limx→∞

{Pr

{wM≥kcx

13

}x−N

} 1N

;

Ξ3 ≥ limx→∞

{Pr

{q≥kcx

13

}x−N

} 1N

= limx→∞

{Pr

{qi≥kcx

13

}x− N

M

}MN

;

Ξ3 ≤ limx→∞

Pr

{w≥ kcx

13

2

}+Pr

{q≥ kcx

13

2

}x−N

1N

.

(70)Thus, by using L’Hopital’s rule, the inequalities in Eq. (70)

can be rewritten as

Ξ3 ≥ limx→∞

1kc3M3

{fw(x)

3Nx−(3N+1)

} 1N

= k−3c M−3k5ϕ21N ;

Ξ3 ≥ limx→∞

1kc3

{fq(x)

3NM x−( 3N

M+1)

}MN

= k−3c k6φ2MN ;

Ξ3 ≤ 8k−3c

(k5Nϕ2 + k6

Nφ2M) 1N

,

(71)



14

where k5 = (3N)−1N and k6 =

(3NM

)−MN . Hence, Ξ3 can bebounded as follows

k−3c θ1N ≤ Ξ3 ≤ 8k−3c (2θ)

1N , (72)

where θ = max{k1Nϕ1, k2Nφ1}, which shows that the upper

and lower bounds of Ξ3 can be determined by ϕ1 and φ1.

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Di Han (S16) received the B.S. degree in in-formation engineering from Shanghai Jiao TongUniversity, Shanghai, China, in 2015, where heis currently pursuing the Ph.D. degree with theDepartment of Electronic Engineering in TsinghuaUniversity, Beijing, China. His research interestsare in wireless communications, with an emphasison mobile computation offloading technologies. Hehas received the Samsung Scholarship in 2013 andNational Scholarship in 2014.



15

Wei Chen (S’05-M’07-SM’13) received his BS andPh.D. degrees (both with the highest honors) fromTsinghua University in 2002 and 2007. From 2005until 2007, he was also a visiting PhD student at theHong Kong University of Science & Technology.Since 2007, he has been on the faculty at TsinghuaUniversity, where he is a tenured full Professor, thedirector of degree office of Tsinghua University, anda university council member. During 2014 to 2016,he served as a deputy head of Department of Elec-tronic Engineering. He visited Princeton University,

Telecom ParisTech, and University of Southampton in 2016, 2014, and 2010respectively. His research interests are in the areas of communication theory,stochastic optimization, and statistical learning.

Dr. Chen is a Cheung Kong young scholar and a member of nationalprogram for special support for eminent professionals, also known as 10,000-talent program. He has also been supported by the national 973 youth project,the NSFC excellent young investigator project, the new century talent programof Ministry of Education, and the Beijing nova program. He received the IEEEMarconi Prize Paper Award and the IEEE Comsoc Asia Pacific Board BestYoung Researcher Award in 2009 and 2011, respectively. He serves as aneditor of the IEEE TRANSACTIONS COMMUNICATIONS and has servedas a TPC co-chair of IEEE VTC-Spring, 2011, as well as, symposium co-chairs for IEEE ICC and Globecom. Dr. Chen is a receipt of the NationalMay 1st Labor Medal and the China Youth May 4th Medal.

Bo Bai received the B.S. degree with the highesthonor in School of Communication Engineeringfrom Xidian University, Xian China, 2004, andthe Ph.D. degree in Department of Electronic En-gineering from Tsinghua University, Beijing Chi-na, 2010. He received the Honor of OutstandingGraduates of Shaanxi Province and the Honor ofYoung Academic Talent of Electronic Engineeringin Tsinghua University. He was a Research Assistantfrom April 2009 to September 2010 and a ResearchAssociate from October 2010 to April 2012 with the

Department of Electronic and Computer Engineering, Hong Kong Universityof Science and Technology. From July 2012 to January 2017, he was anAssistant Professor with the Department of Electronic Engineering, TsinghuaUniversity. He has obtained the support from Backbone Talents SupportingProject of Tsinghua University. Currently, he is a Senior Researcher at TheoryLab, 2012 Labs, Huawei Technologies Co., Ltd., Hong Kong. He is leadinga team to develop fundamental principles, algorithms, and systems for graphinformatics, B5G/6G, and network information theory.

He is an IEEE Senior Member, USENIX Member, and has authored about100 papers in major IEEE/ACM journals and conferences, 2 book chapters,and 1 textbook. He is one of the founded vice chairs of IEEE TCCN SIG onSocial Behavior Driven Cognitive Radio Networks. He served as a committeemember in IEEE ComSoc WTC and IEEE ComSoc SPCE. He served as a TPCco-chair of IEEE Infocom 2018 - 1st AoI Workshop and IEEE Infocom 2019- 2nd AoI Workshop, a TPC co-chair of IEEE ICCC 2018, and an IndustrialForum & Exhibition co-chair of IEEE HotICN 2018. He also served as a TPCmember for several IEEE conferences such as ICC, Globecom, WCNC, VTC,and ICCC. He served as a reviewer for several major IEEE/ACM journalsand conferences. He was the recipient of the Student Travel Grant at IEEEGlobecom 2009. He was invited as a Young Scientist Speaker at IEEE TTM2011. He was a recipient of the Best Paper Award in IEEE ICC 2016.

Yuguang Fang (F’08) received an MS degree fromQufu Normal University, Shandong, China in 1987,a PhD degree from Case Western Reserve Universityin 1994, and a PhD degree from Boston Universityin 1997. He joined the Department of Electrical andComputer Engineering at University of Florida in2000 and has been a full professor since 2005. Heholds a University of Florida Research Foundation(UFRF) Professorship (2017-2020, 2006-2009), Uni-versity of Florida Term Professorship (2017-2019),a Changjiang Scholar Chair Professorship (Xidian

University, Xian, China, 2008-2011; Dalian Maritime University, Dalian,China, 2015-2018), Overseas Adviser, School of Information Science andTechnology, Southwest Jiao Tong University, Chengdu, China (2014-present),and Overseas Academic Master (Dalian University of Technology, Dalian,China, 2016-2018).

Dr. Fang received the US National Science Foundation Career Awardin 2001, the Office of Naval Research Young Investigator Award in 2002,the 2018 IEEE Vehicular Technology Outstanding Service Award, the 2015IEEE Communications Society CISTC Technical Recognition Award, the2014 IEEE Communications Society WTC Recognition Award, and the BestPaper Award from IEEE ICNP (2006). He has also received a 2010-2011UF Doctoral Dissertation Advisor/Mentoring Award, a 2011 Florida BlueKey/UF Homecoming Distinguished Faculty Award, and the 2009 UF Collegeof Engineering Faculty Mentoring Award. He was the Editor-in-Chief of IEEETransactions on Vehicular Technology (2013-2017), the Editor-in-Chief ofIEEE Wireless Communications (2009-2012), and serves/served on severaleditorial boards of journals including Proceedings of the IEEE (2018-present),ACM Computing Surveys (2017-present), IEEE Transactions on MobileComputing (2003-2008, 2011-2016), IEEE Transactions on Communications(2000-2011), and IEEE Transactions on Wireless Communications (2002-2009). He has been actively participating in conference organizations suchas serving as the Technical Program Co-Chair for IEEE INFOCOM2014 andthe Technical Program Vice-Chair for IEEE INFOCOM’2005. He is a fellowof the IEEE and a fellow of the American Association for the Advancementof Science (AAAS).

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